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Mirrors > Home > MPE Home > Th. List > prel12g | Structured version Visualization version Unicode version |
Description: Closed form of prel12 4383. (Contributed by AV, 9-Dec-2018.) |
Ref | Expression |
---|---|
prel12g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2626 | . . . . . . 7 | |
2 | 1 | notbid 308 | . . . . . 6 |
3 | preq1 4268 | . . . . . . . 8 | |
4 | 3 | eqeq1d 2624 | . . . . . . 7 |
5 | eleq1 2689 | . . . . . . . 8 | |
6 | 5 | anbi1d 741 | . . . . . . 7 |
7 | 4, 6 | bibi12d 335 | . . . . . 6 |
8 | 2, 7 | imbi12d 334 | . . . . 5 |
9 | 8 | imbi2d 330 | . . . 4 |
10 | eqeq2 2633 | . . . . . . 7 | |
11 | 10 | notbid 308 | . . . . . 6 |
12 | preq2 4269 | . . . . . . . 8 | |
13 | 12 | eqeq1d 2624 | . . . . . . 7 |
14 | eleq1 2689 | . . . . . . . 8 | |
15 | 14 | anbi2d 740 | . . . . . . 7 |
16 | 13, 15 | bibi12d 335 | . . . . . 6 |
17 | 11, 16 | imbi12d 334 | . . . . 5 |
18 | 17 | imbi2d 330 | . . . 4 |
19 | preq1 4268 | . . . . . . . 8 | |
20 | 19 | eqeq2d 2632 | . . . . . . 7 |
21 | 19 | eleq2d 2687 | . . . . . . . 8 |
22 | 19 | eleq2d 2687 | . . . . . . . 8 |
23 | 21, 22 | anbi12d 747 | . . . . . . 7 |
24 | 20, 23 | bibi12d 335 | . . . . . 6 |
25 | 24 | imbi2d 330 | . . . . 5 |
26 | 25 | imbi2d 330 | . . . 4 |
27 | preq2 4269 | . . . . . . . . 9 | |
28 | 27 | eqeq2d 2632 | . . . . . . . 8 |
29 | 27 | eleq2d 2687 | . . . . . . . . 9 |
30 | 27 | eleq2d 2687 | . . . . . . . . 9 |
31 | 29, 30 | anbi12d 747 | . . . . . . . 8 |
32 | 28, 31 | bibi12d 335 | . . . . . . 7 |
33 | 32 | imbi2d 330 | . . . . . 6 |
34 | vex 3203 | . . . . . . 7 | |
35 | vex 3203 | . . . . . . 7 | |
36 | vex 3203 | . . . . . . 7 | |
37 | vex 3203 | . . . . . . 7 | |
38 | 34, 35, 36, 37 | prel12 4383 | . . . . . 6 |
39 | 33, 38 | vtoclg 3266 | . . . . 5 |
40 | 39 | a1i 11 | . . . 4 |
41 | 9, 18, 26, 40 | vtocl3ga 3276 | . . 3 |
42 | 41 | 3expa 1265 | . 2 |
43 | 42 | impr 649 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 |
This theorem is referenced by: hash2prd 13257 |
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